Bayes' Rule

If the set of events A₁ , A₂ , ....., A_n constitutes a partition of the sample space S, and event B is a subset of S, then

B = B Ç S

= B Ç (A₁ È A₂ È ......... È A_n )
= (B Ç A₁ ) È (B Ç A₂ ) È ....... È (B Ç A_n)

As the events A₁ , A₂ , ......, A_n are mutually exclusive, then the events (B Ç A_i ), where i ÃŽ{1, 2, ...., n }, is also mutually exclusive. Assuming that none of the events A₁ , A₂ ,....., A_n is null, i.e. P(A_i ) ¹ 0 , i ÃŽ{1, 2,...., n }

P(B) = P(B Ç A₁ ) + P(B Ç A₂ ) + ....... + P(B ÇA_n)

= P(A₁ ) P(B | A₁ ) + P(A₂) P(B | A₂) + ....... + P(A_n ) P(B | A_n)

From the definition of conditional probability,

Thus we have derived Bayes' Rule, which states the following:

Bayes' Rule. If the set of events A₁ , A₂ ,....., A_n constitutes a partition of the sample space S, P(A_i ) ¹ 0 , i ÃŽ{1, 2,....., n }, and event B is a subset of S, P(B) ¹ 0,

EX. A family had plans to go fishing on a Sunday afternoon, but their plans were dependent on the weather at noon Sunday. If it was sunny, then there was a 90 % chance that they would go fishing. If it was cloudy, then the probability that they would go fishing would drop to 50 %. And if it was raining, the chances dropped to 15 %. The weather prediction, which we can assume to be accurate, called for a 10 % chance of rain, a 25 % chance of clouds, and a 65 % chance of sunshine.

Set events

F as the event that the family goes fishing
S as the event that the weather is sunny at Sunday noon
C as the event that the weather is cloudy at Sunday noon
R as the event that the weather is rainy at Sunday noon

P(S) = 0.65, P(C) = 0.25, P(R) = 0.10
Note that P(S) + P(C) + P(R) = 1, and of course S, C and R are mutually exclusive events.

P(F|S) = 0.90, P(F|C) = 0.50, P(F|R) = 0.15
P(F) = P(F|S) P(S) + P(F|C) P(C) + P(F|R) P(R)
= (0.90)(0.65) + (0.50)(0.25) + (0.15)(0.10)
= 0.585 + 0.125 + 0.015
= 0.725

Assuming that the family ends up going fishing, the probability of each type of weather occurring is

P(S|F) = probability of sunny weather, given that the family went fishing.

P(S|F) = probability of cloudy weather, given that the family went fishing.

P(S|F) = probability of rainy weather, given that the family went fishing.

Note that P(S|F) + P(C|F) + P(R|F) = 0.807 + 0.172 + 0.021 = 1.000